It does, here's a great video by Ben at Applied Science showing some examples of the effects. https://www.youtube.com/watch?v=JV4Fk3VNZqs

The best lightsaber theory I've read is how lightsabers cut through everything but not another lightsaber. It's by magnetic repulse force of course.

Magnet does affect flame, but you need surprisingly strong magnet for the effect to be noticeable.